Should you eat back your exercise calories burned?

yes -that is the way MFP is designed to be used.

Probably.

If you got your base calorie estimate from MFP, and didn't include regular bike rides when setting your activity level, then yes: Add and eat back a reasonable estimate of the exercise that isn't part of your normal activity level. MFP's design intention is that we set activity level based on pre-exercise activity, then add exercise when we do it, and eat those calories, too.

But some people get a calorie estimate from an outside TDEE calculator, and include their exercise plans in their activity level (or do similarly when setting up MFP). If a person has already considered exercise plans, and included them in activity level, then adding exercise on top of that would be double-counting exercise calories.

Tom is correct: It's important to evaluate results after 4-6 weeks (or one whole menstrual cycle for those who have them), then adjust if necessary. (Not every one of us is statistically average, though most of us are close.)

Either of the above methods can work fine, if a person is clear-headed about the assumptions built into their calorie estimates, and behaving accordingly. Of course it depends on reasonably on-point estimates for base calories and exercise, but that's what the evaluation phase is for.

Best wishes!

P.S. I estimated my exercise calories carefully, and ate pretty much all of them through about a year of weight loss, and 8 years of maintaining a healthy weight since. It has worked fine, for me . . . after that "assess then adjust" few weeks up front.

There are only 2 bad choice extremes available here: One is double-counting exercise (including it in activity level, then adding it on also). The other is setting a very aggressive weight loss rate, then doing a bunch of exercise and making weight loss happen even faster. Very fast weight loss increases health risks, and makes the process hard to stick with long enough to lose a meaningful total amount of weight. In between those extremes, it's a matter of personal risk tolerance. (I don't like risking my health, personally.)

For eating back calories to form an accurate picture of how successful you’ll be you have several factors that need to be accurate.
-proper activity level choice
-accurate weekly calorie counting and tracking
-choosing the correct data entries of foods/drinks
-not overestimating calories burned

So you can see how it becomes difficult to get it right. Stick to a plan for a month or so and adjust accordingly.